Strategies for picking good colour palettes? [migrated]

I'm new to web design. I learned the basic stuff like layout, contrasts, the difference between RGB and CMYK, etc., but I struggle to apply this well in practice:

Picking good colours that work well with one another.

For instance, say I choose a pantone red for the logo, a near-black for text and a simple grey for borders.

What would a seasoned web designer do if they wanted to introduce new colours into the web page to highlight a button, while making sure that the new colour fits well into the current palette?